Heat Transfer. Revision Examples

Transcription

1 Hea Transfer Revision Examples

2 Hea ransfer: energy ranspor because of a emperaure difference. Thermal energy is ransferred from one region o anoher. Hea ranspor is he same phenomena lie mass ransfer, momenum ransfer and elecrical conducion. Similar rae equaions, where flux is proporional o a poenial difference. Poenial difference emperaure difference Three modes of hea ransfer: - Thermal Conducion - Thermal Convecion - Thermal Radiaion

3 Thermal conducion The mechanism: energy is ranspored beween pars of coninuum by he ransfer of ineic energy beween paricles or groups of paricles a he aomic level. Purely hermal conducion: in solid opaque bodies (opaque: no permeable for radiaion) he hermal conducion is he significan hea ransfer mechanism because he maerial doesn flow and here is no radiaion. In flowing fluids, hermal conducion dominaes in he region very close o he boundary layer, where he flow is laminar he flow parallel o he surface here is no eddy moion 3

4 Seady-sae conducion Seady-sae condiions: = F() = consan For seady-sae hea conducion, in one dimension, he Fourier-law is q ( A) d dx q hea flow rae, W hermal conduciviy or hea conducion coefficien, W/m K A cross secional area normal o flow, m d/dx emperaure gradien, K/m The equaion saes ha he hea flow rae (q) in he x direcion is direcly proporional o he emperaure gradien d/dx and he cross secional area A normal o he hea flow. The proporionaliy facor is he hermal conduciviy. The minus sign indicaes ha he hea flow is posiive in he direcion of decreasing emperaure. 4

5 If A and are consan (e.g. a a wall) he equaion q A ( ) L L wall hicness emperaure a x = 0 emperaure a x = L A surface of he wall hermal conduciviy 5

6 One-layer fla wall W, x x A x x A q L R R A R A L A L A q where W, R hermal resisance, K/W 6

7 Example Given is a solid bric wall, wih he following daa: Maerial: bric, fired clay, densiy: 760 g/m 3, conduciviy (): 0.8 W/m K Surface (A): 5 m x 3 m = 5 m Inerior surface emperaure ( ): 0 C Exerior surface emperaure ( ): 0 C Thicness of he wall (L): 38 cm = 0.38 m Calculae a) he hea flow hrough he wall b) he hermal resisance 7

8 Muli-layer fla wall W, R A R A R A q If he number of layers is n: n i i n R A q 8

9 Example Given is an insulaed concree wall, which daa are: Layers: sand and gravel concree, 400 g/m 3, =.5 W/m K, L = 6 cm expanded polysyrene, 0 g/m 3, = W/m K, L = 6 cm 3 sand and gravel concree, 400 g/m 3, 3 =.5 W/m K, L 3 = 8 cm Surface (A): 5 m x 3 m = 5 m Inerior surface emperaure ( ): 0 C Exerior surface emperaure ( 4 ): 0 C a) Calculae he hea flow hrough he wall and he emperaure on he surface of he layers b) Illusrae he emperaure in he wall 9

10 Thermal conducion a a one-layer pipe The Fourier-equaion in cylinder coordinaes: Q r L d dr, W q r L d/dr he hea flow rae radius lengh of he pipe hermal conduciviy emperaure gradien 0

11 The soluion of he equaion: W, r r ln L Q W, R Q The hermal conducion resisance: W K m, ln or ln d d L R r r L R A he same way, wih he hermal conducion resisance:

12 Thermal convecion Energy ransfer is involved by fluid movemen and molecular conducion. Hea ransfer means energy ransfer from liquids and gases o he surface of a body or a wall, or from he surface of a body o he liquid.

13 If in he flow he Reynolds number is large enough, hree differen flow regions exis: A he wall: laminar sublayer (boundary layer) hermal conducion Ouside he laminar layer: buffer layer eddy mixing and conducion Beyond he buffer layer: urbulen region eddy mixing 3

14 Convecion is divided ino wo caegories: Free convecion, naural convecion: he flow is generaed by nonhomogeneous densiies caused emperaure difference. Forced convecion: he flow is produced by exernal sources (pump, fan). 4

15 The hea flow rae is described by he Newon s formula: q h A( ) c s f h c A Where q hea flow rae, W h c convecional hea ransfer coefficien, W/m K A surface of he wall, m s emperaure of he surface, K or C f emperaure of he fluid, K or C 5

16 The hea ransfer coefficien can be calculaed wih dimensionless numbers, from similariy heory. Generally Nu = F (Pr; Gr; Re) I means, Nussel number is a funcion of he produc of Prandl number, Grashof number and Reynolds number. Dimensionless Numbers h c L Nu h c hea ransfer coefficien, W/m K L characerisic dimension, m hermal conduciviy of he fluid, W/m K 6

17 Dimensionless Numbers In he equaion of Nussel number Pr c p absolue viscosiy, g/m s or Ns/m c p specific hea, J/g K hermal conduciviy of he fluid, W/m K Gr L 3 g L characerisic dimension, m densiy, g/m 3 coefficien of hermal expansion, /K g graviaional acceleraion, m/s emperaure difference, K or C Re v L absolue viscosiy, g/m s or Ns/m v velociy, m/s L characerisic dimension, m D, diameer inemaic viscosiy = /, m /s 7

18 Pracical cases of convecion Naural or free convecion: effec of emperaure difference. General relaionship: n c Gr C L h Nu Pr) ( n f p n f c c g L L C h av 3 8

19 Open spaces: h c = F(; direcion of hea flow) Direcion of hea flow: Ceiling down Wall Floor horizonal up = s air > 0 h c up >h c hor > h c down Some values for h c hea ransfer coefficien Air, gas 4 0 W/m K Waer, liquid W/m K 9

20 Forced convecion: fluid sream derived from ouer force Nu = F(Pr, Re) = F(v) Some values for h c hea ransfer coefficien Air, gas 0 50 W/m K Waer, liquid W/m K 0

21 Example 3 Given are a room and a radiaor. Surface emperaure: s = 85 C Parameers of ambien air: air = 30 C, bar, dry air Naural convecion, lengh in direcion of flow: 0,5 m

22 Soluion: Nu h c L C ( Gr Pr) n h c C L L 3 g av n f c p n f Properies of dry air a air = 30 C densiy, 30 =.70 g/m 3 coefficien of hermal expansion, 30 = /K graviaional acceleraion, g = m/s absolue viscosiy, 30 = Ns/m specific hea, c p30 = 03 J/g K hermal conduciviy of he fluid, 30 = W/m K Temperaure difference = = 55 C Average emperaure av = ( )/ = 5/ ~ 60 C absolue viscosiy a 60 C 60 = Ns/m

23 ASHRAE Fundamenals 005, Chaper 3, Page 3.7 Table 0 3

24 A special problem of convecion: Convecion in closed spaces: hollows, air layer beween surfaces e.g. window consrucion > A = A Complex process of hea ransfer: conducion convecion radiaion Equivalen hea conducion coefficien: e Equivalen conducion resisance of air layer: R equi L equi From able, e.g. ASHRAE Fundamenals 00, Chaper 5, Table 3 Thermal Resisances of Plane Air spaces, m K/W 4

25 Thermal radiaion The radiaion energy ransfer is hrough energy-carrying elecromagneic waves ha are emied by aoms and molecules due o change in heir energy conen. I means: does no depend on an inermediae maerial. The rae of hermal energy emied by a surface depends on is quaniy and is absolue emperaure. A blac surface absorbs all inciden radiaion. 5

26 The oal energy emied per uni ime and uni area is given by he Sefan-Bolzman law: E blac T 4 where Sefan-Bolzmann consan: W/m K 4 For nonblac surfaces E Eblac T 4 where hemispherical emiance or emissiviy. is a funcion of he maerial, condiion of is surface. 6

27 Lamber s cosine law: Lamber's cosine law is he saemen ha he oal power observed from a "Lamberian" surface is direcly proporional o he cosine of he angle Φ made by he observer's line of sigh and he line normal o he surface. E E n cos Uilising he Lamber s law he oal energy radiaed o he hemisphere is: E Toal E n 7

28 The hea flow rae beween he surfaces Given are wo surfaces: 8

29 Firs le s examine wo general plane elemens! The energy flux which leaves da elemen and which is absorbed by da elemen is quadraicly small: d E = E n cos da dw From his, he radiaion absorbed by da is: d E - = d E The hea exchange hrough radiaion beween he wo plane elemens is: d q - = d E - d E - 9

30 Angle Facor The fracion of all radian energy leaving a surface i ha is direcly inciden on surface is he angle facor F i (also nown as view facor, shape facor, and configuraion facor). 30

31 3

32 3

33 Overall Hea Transfer From inside air o ouside air he hea ransfer is a complex process: Inside convecion In he wall consrucion conducion Ouside also convecion The hea flow rae wih overall hea ransfer coefficien: q U A ( ) i o Overall hea ransfer coefficien: U h i n j L i i h o or U R is n j R j R os 33

34 Example 4 Le s see Example There is a bric wall, where he wall consrucion daa are he following: Maerial Densiy Conduiviy Thicness, g/m 3, W/m,K L, m Bric, fired clay R = m K/W Inside emperaure: C Ouside emperaure: 4 C Area of he surface: 5 m 34

35 Hea ransfer coeff./surface resisance on he laminar air layer W/m K ASHRAE F.5. EN 83 Sill air (inside) Horizonal surface (upward) 9.6/0.* 0.0/0. Horizonal surface (downward) 6.3/0.6* 5.9/0.7 Verical surface 8.9/0.* 7.7/0.3 * non reflecive surfaces Moving air (ouside) Any posiion 34/0.03* 5/0.04 * wind velociy 6,7 m/s (4 m/h) 35

36 Hungary: 7/006. Miniserial Decree energeical calculaions regulae he energy consumpion of buildings in harmony wih he european regulaions. Required U-values: Exernal wall.. Ec W/m K 36

37 Example 5 We improve he bric wall wih a new insulaion The wall consrucion daa are he following: Maerial Densiy Conduiviy Thicness, g/m 3, W/m,K L, m Bric, fired clay Expanded polysyrene R = m K/W Inside emperaure: C Ouside emperaure: 4 C Area of he surface: 5 m 37